How to Construct a Prior Probability Distribution From a Training Image?

Abstract:

Probabilistic inverse problems and geostatistical modeling rely on trustworthy prior information. The choice of prior distribution may have a considerably impact on uncertainty quantification for such modeling problems. For these problems, prior information is expressed by a probability distribution over the parameters describing the earth system. In practice, prior information is often given as incomplete knowledge extracted from one or a few realizations (e.g., training images) from the assumed underlying stochastic process describing the subsurface. From such information, approximate marginal distributions from this unknown underlying process can be derived (by assuming the underlying process to be stationary) in form of e.g. semivariograms (i.e., 2D marginals) or multiple-point-based pattern histograms (i.e., >2D marginals). We discuss how to determine a joint probability distribution that describes this underlying process (i.e., a prior probability distribution) that is consistent with the known marginal distributions obtained from the training image(s). It turns out that the general problem of determining a joint probability distribution that is consistent with such known marginal distributions is underdetermined. Consequently, infinitely many possible joint (prior) distributions exist that that are consistent with these marginals. We investigate some sampling algorithms, which sample prior distributions based on training image statistics, and find that they do not produce samples that are consistent with training image (marginal) statistics. Finally, we show examples of using subjective chosen constraints, such as the Gaussian assumption or Markov properties, to supplement the marginals in order to build a well-defined prior distribution.